**STATEMENT**
Bernoulli's theorem states that when a liquid is flowing, the total of the pressure energy, kinetic energy and potential energy per unit mass should be constant.

**EQUATION OF BERNOULLI'S THEOREM**
The equation is

+

+gh=constant.

**ASSUMPTION**
Let us assume a streamline flow of liquid which has a density

.It passes through a pipe XY. Assume that the pressures are denoted as P1 and P2. The cross sectional areas of the pipe are a1 and a2. These cross sectional areas are at the point X and Y. Let v1 be the velocity of the liquid which enters the pipe through X and v2 be the velocity of the liquid which leaves the pipe through Y. In this case when the height of the Y is greater than X, the liquid is accelerated. This acceleration should be forced against the force of gravity. Hence the pressure P1 is greater than the pressure P2

**DERIVATION OF BERNOULLI PRINCIPLE:**
Consider the mass m which is to be passed into the pipe, then

a

_{1}v

_{1}=a

_{2}v

_{2}=m

Dividing the equation by

we get,

a

_{1}v

_{1}=a

_{2}v

_{2}=

=V

Force acting on liquid at X = p

_{1}a

_{1}
Force acting on the liquid at Y = p

_{2}a

_{2}
Work done per second at X = p

_{1}a

_{1}X v

_{1}=p

_{1}V

Work done per second at Y= p

_{2}a

_{2}XW

_{2}=p

_{2}V

Total work performed per second on the liquid by the pressure starting from X to Y =p

_{1}V-p

_{2}V

If the mass of the liquid is m, then the potential energy per second on the liquid from X to Y is equal to mgh

_{2}-mgh

_{1}
Now the increase of kinetic energy per second=

mv

_{2}^{2}-

mv

_{1}^{2}
The work energy principle states that the work done per second by the pressure is equal to adding the increase in potential energy per second with the increase in kinetic energy per second.

P

_{1}V-P

_{2}V=(mgh

_{2}-mgh

_{1})+

P

_{1}V+mgh

_{1}+

mv

_{1}^{2}= P2V+mgh2+

mv2

^{2} On dividing by m throughtout,

+gh1+

v1

^{2}=

+gh2+

v

_{2}2

+gh1+

v12=

+gh2+

v2

^{2}
+gh+

v

^{2}=constant

When dividing by g, the term becomes

+

+h=constant

Here the terms

,

,h
are called pressure head,velocity head and gravitational head respectively.

Hence the

*Daniel Bernoulli's principle* theorem is proved..